Selected Publications by Topic

I. Hyperbolic dynamical systems: general theory


Entropy, Lyapunov exponents & dimension

• Dimension, entropy and Lyapunov exponents, Ergod. Thy & Dynam. Sys., 2, (1982), 109-124.

• (with F. Ledrappier) The metric entropy of diffeomorphisms, Part I: Characterization of measures satisfying Pesin's entropy formula, Annals Math., 122, (1985), 509-539.

• (with F. Ledrappier) The metric entropy of diffeomorphisms, Part II: Relations between entropy, exponents and dimension, Annals Math., 122, (1985), 540-574.

• (with F. Ledrappier) Entropy formula for random transformations, Prob. Th. Rel. Fields, 80, (1988), 217-240.

• (with F. Ledrappier) Dimension formula for random transformations, Commun. Math. Phys., 117, (1988), 529-548.

SRB measures

• Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. AMS, 287, (1985), 41-48.

• (with F. Ledrappier) The metric entropy of diffeomorphisms, Part I: Characterization of measures satisfying Pesin's entropy formula, Annals Math., 122, (1985), 509-539.

• (with F. Ledrappier) Entropy formula for random transformations, Prob. Th. Rel. Fields, 80, (1988), 217-240.

• (with H. Hu) Nonexistence of SBR measures for some diffeomorphisms that are ``almost Anosov", Ergod. Th. & Dynam. Syst., 15, (1995), 67-76.
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• Statistical properties of dynamical systems with some hyperbolicity, Annals of Math., (1998), 585-650.
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• What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., vol. 108, Issue 5, (2002), 733-754.
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  (with W. Cowieson) SRB measures as zero-noise limits, Ergod. Th. & Dynam. Sys., Vol. 25, 4, (2005) 1091-1113.
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Randomly perturbed systems

• Stochastic stability of hyperbolic attractors, Ergod. Th. & Dynam. Sys., 6, (1986), 311-319.

• Random perturbations of matrix cocycles, Ergod. Th. & Dynam. Sys., 6, (1986), 627-637.

• (with F. Ledrappier) Entropy formula for random transformations, Prob. Th. Rel. Fields, 80, (1988), 217-240.

• (with F. Ledrappier) Dimension formula for random transformations, Commun. Math. Phys., 117, (1988), 529-548.

• (with F. Ledrappier) Stability of Lyapunov exponents, Ergod. Th. & Dynam. Sys., 11, (1991), 469-484.

• (with V. Baladi) On the spectra of randomly perturbed expanding maps, Commun. Math. Phys., 156, (1993), 355-385.
  [ PDF | Postscript ] Erratum: [ PDF | Postscript ]

  (with W. Cowieson) SRB measures as zero-noise limits, Ergod. Th. & Dynam. Sys., Vol. 25, 4, (2005) 1091-1113.
  [ PDF | Postscript ]

Lyapunov exponents

• Random perturbations of matrix cocycles, Ergod. Th. & Dynam. Sys., 6, (1986), 627-637.

• (with F. Ledrappier) Stability of Lyapunov exponents, Ergod. Th. & Dynam. Sys., 11, (1991), 469-484.

• Some open sets of nonuniformly hyperbolic cocycles, Ergod. Th. & Dynam. Sys., 13, (1993), 409-415.
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• Lyapunov exponents for some quasi-periodic cocycles, Ergod. Th. & Dynam. Syst., 17, (1997), 483-504
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Statistical properties

• Some large deviation results for dynamical systems, Trans. AMS, 318, (1990), 525-543.

• Statistical properties of dynamical systems with some hyperbolicity, Annals of Math., (1998), 585-650
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• Recurrence times and rates of mixing, Israel J Math. 110, (1999), 153-188
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• (with E. Kobre) Extended Systems with Deterministic Local Dynamics and Random Jumps, to appear in Commun. Math. Phys.
  

• (with L. Rey-Bellet) Large Deviations in nonuniformly hyperbolic dynamical systems, to appear in Erg. Th. & Dynam. Sys.
  

• (with W. Ott and M. Stenlund) Memory loss for time-dependent dynamical systems, to appear in Math Research Letters
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Strange attractors

• Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. AMS, 287, (1985), 41-48.

• (with Q.D. Wang) Strange attractors with one direction of instability, Commun. Math. Phys. 218, (2001), 1--97.
  [ PDF | Postscript ]

• What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., vol. 108, Issue 5, (2002), 733-754.
  [ PDF | Postscript ]

• (with Q.D. Wang) Toward a theory of rank one attractors, Annals of Math, Vol 167. No.2 (2008) 349-480.
  [ PDF | Postscript ]

II. Analysis of concrete systems


Billiards

• Statistical properties of dynamical systems with some hyperbolicity, Annals of Math., (1998), 585-650
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• (with N. Chernov) Decay of correlations for Lorentz gases and hard balls, Encycl. of Math. Sc., Math. Phys. II, Vol. 101, Ed. Szasz, (2001), 89-120.
  [ PDF | Postscript ]

• (with M. Demers and P. Wright) Escape rates and physically relevant measures for billiards with small holes, to appear
  

1-d maps

• (with L. Block, J. Guckenheimer and M. Misiurewicz) Periodic points and topological entropy of one-dimensional maps, Springer Lecture Notes in Math., 819, (1980), 18-34.

• (with P. Thieullen and C. Tresser) Exposant de Lyapunov positif dans des familles a un parametre d'applications unimodales, C.R. Acad. Sci. Paris, 315.I, (1992), 69-72; J d'Analyse, 64, (1994), 121-172.

• (with M. Benedicks) Absolutely continuous invariant measures and random perturbations of certain one-dimensional maps, Ergod. Th. & Dynam. Sys, 12, (1992), 13-37.
  

• Decay of correlations for certain quadratic maps, Commun. Math. Phys, 146, (1992), 123-138.
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• (with Q.D. Wang) Nonuniformly expanding 1D maps, Commun. Math. Phys. Vol. 264, No.1 (2006) 255-282.
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Henon attractors

• (with M. Benedicks) Sinai-Bowen-Ruelle measures for certain Henon maps, Invent. Math, 112, (1993), 541-576.
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• (with M. Benedicks) Markov extensions and decay of correlations for certain Henon maps, Asterisque 261, (2000), 13-56.
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Rank-one attractors

• (with Q.D. Wang) Strange attractors with one direction of instability, Commun. Math. Phys. 218, (2001), 1--97.
  [ PDF | Postscript ]

• (with Q.D. Wang) From invariant curves to strange attractors, Commun. Math. Phys. 225, (2002), 275-304.
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• (with Q.D. Wang) Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Commun. Math. Phys. 240, (2003), 509-529.
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• (with Q.D. Wang) Toward a theory of rank one attractors, Annals of Math, Vol 16 7. No.2 (2008) 349-480.
  [ PDF | Postscript ]

Other systems

• (with I. Klapper) Rigorous bounds on the fast dynamo growth rate involving topological entropy, Commun. Math Phys., 173, (1995), 623-646.
  [ PDF | Postscript ]

• (with N. Masmoudi) Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs, Commun. Math. Phys., 227, (2002), 461-481.
  [ PDF | Postscript ]

• (with J. Guckenheimer and M. Weschelberger) Chaotic attractors of relaxation oscillators, Nonlinearity 19, No. 3 (2006) 701-720.
  [ PDF | Postscript ]

III. Applications of dynamical systems ideas


Stochastic behavior of deterministic systems

• Statistical properties of dynamical systems with some hyperbolicity, Annals of Math., (1998), 585-650.
  [ PDF | Postscript ]

• Recurrence times and rates of mixing, Israel J Math. 110, (1999), 153-188.
  [ PDF | Postscript ]

• (with E. Kobre) Extended Systems with Deterministic Local Dynamics and Random Jumps, to appear in Commun. Math. Phys.
  

• (with L. Rey-Bellet) Large Deviations in nonuniformly hyperbolic dynamical systems, to appear in Erg. Th. \& Dynam. Sys.
  

Detecting chaotic behavior

• (with Q.D. Wang) From invariant curves to strange attractors, Commun. Math. Phys. 225, (2002), 275-304.
  [ PDF | Postscript ]

• (with Q.D. Wang) Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Commun. Math. Phys. 240, (2003), 509-529.
  [ PDF | Postscript ]

• (with J. Guckenheimer and M. Weschelberger) Chaotic attractors of relaxation os cillators, Nonlinearity 19, No. 3 (2006) 701-720.
  [ PDF | Postscript ]

• (with K. Lin) Shear-induced chaos, 2007 preprint
  

Nonequilibrium statistical mechanics

• (with J.-P. Eckmann) Temperature profiles in Hamiltonian heat conduction, Europhysics Letters, Volume 68, Issue 6, December 2004, 790-796.
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• (with J.-P. Eckmann) Nonequilibrium energy profiles for a class of 1-D models, Commun. Math. Phys. Vol. 262, No. 1 (2006) 237-267.
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• (with K. Lin) Correlations in Nonequilibrium Steady States of Random Halves Models, J. Stat. Phys., Vol. 128, No. 3, (2007), 607-639.
  

• (with K. Ravishankar) Local Thermodynamic Equilibrium for some Stochastic Models of Hamiltonian Origin, J Stat. Phys., Vol. 128, No. 3, (2007), 641-665.
  

• (with E. Kobre) Extended Systems with Deterministic Local Dynamics and Random Jumps, to appear in Commun. Math. Phys.
  

• (with P. Balint and K. Lin) Ergodicity and energy distributions for some boundary driven integrable Hamiltonian chains, to appear
  

Models from Physics & Biology

• (with I. Klapper) Rigorous bounds on the fast dynamo growth rate involving topological entropy, Commun. Math Phys., 173, (1995), 623-646.
  [ PDF | Postscript ]

• (with N. Chernov) Decay of correlations for Lorentz gases and hard balls, Encycl. of Math. Sc., Math. Phys. II, Vol. 101, Ed. Szasz, (2001), 89-120.
  [ PDF | Postscript ]

• (with N. Masmoudi) Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs, Commun. Math. Phys., 227, (2002), 461-481.
  [ PDF | Postscript ]

• (with J. Guckenheimer and M. Weschelberger) Chaotic attractors of relaxation os cillators, Nonlinearity 19, No. 3 (2006) 701-720.
  [ PDF | Postscript ]

• (with K. Lin and E. Shea-Brown) Reliability of coupled oscillators, to appear in J Nonlinear Science
  

• (with K. Lin and E. Shea-Brown) Spike-Time Reliability of Layered Neural Oscillator Networks, J Comp Neurosci
  

IV. Expository articles


General

• Ergodic theory of chaotic dynamical systems, Proc. of SMALEFEST, Springer-Verlag, (1993), 201-226.
  

• Ergodic theory of chaotic dynamical systems, Proc. International Congress of Math. Phys., Brisbane 1997, International Press (1999), 132-143.
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• Developments in chaotic dynamics, AMS Notices, Nov. 1998.
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• Geometric and ergodic theory of hyperbolic dynamical systems, Lecture series in Current Developments in Mathematics 1998, Harvard-MIT, International Press (1999).
  [ PDF | Postscript ]

• Chaotic phenomena in three settings: large, noisy, and out of equilibrium, to appear in Nonlinearity (invited article "Open Problems")
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Reviews of specific topics

• Dimension, entropy and Lyapunov exponents in differentiable dynamical systems, Physica A, 124, (1984), 639-646 (Proceedings of ICMP Boulder 1983)

• (with N. Chernov) Decay of correlations for Lorentz gases and hard balls, Encycl. of Math. Sc., Math. Phys. II, Vol. 101, Ed. Szasz, (2001), 89-120.
  [ PDF | Postscript ]

• Entropy in dynamical systems, in Entropy, Ed. Greven Keller and Warnecke, Princeton Univ. Press, (2003), 313-328.
  [ PDF | Postscript ]

• What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., vol. 108, Issue 5, (2002), 733-754.
  [ PDF | Postscript ]

• (with Q.D. Wang) Nonuniformly expanding 1D maps, Commun. Math. Phys. Vol. 264, No.1 (2006) 255-282.
  [ PDF | Postscript ]

• (with M. Demers) Escape rates and conditionally invariant measures, Nonlinearity 19 (2006) 377-397.
  [ PDF | Postscript ]

Lecture Notes

• Ergodic theory of differentiable dynamical systems, "Real and Complex Dynamics", Ed. Branner and Hjorth, NATO ASI series, Kluwer Academic Publishers, (1995), 293-336.
  [ PDF | Postscript ]

[ General hyperbolic theory ]
[ Analysis of specific systems ]
[ Applications of dynamical systems ideas ]
[ Expository articles ]